Revista Colombiana de Matemáticas
Volumen 50(2016)2, páginas 165-174
Construction of Bh[g] sets in product of
groups
Construcción de conjuntos Bh [g] en producto de grupos
Diego Ruiz1,B , Carlos Trujillo1
1
Universidad del Cauca, Popayán, Colombia
Abstract. A subset A of an abelian group G is a Bh [g] set on G if the elements
of G can be written in at most g ways as sum of h elements of A. Given any
field F, this work
presents constructions of Bh [g] sets on the abelian groups
Fh , + , Zd , + , and (Zm1 × · · · × Zmd , +), for d ≥ 2, h ≥ 2, and g ≥ 1.
Key words and phrases. Sidon sets, Bh [g] sets.
2010 Mathematics Subject Classification. 11B50, 11B75.
Resumen. Un subconjunto A de un grupo abeliano G es un conjunto Bh [g]
sobre G si todo elemento de G puede escribirse en a lo sumo de g formas como la suma de h elementos de A. En este trabajo se presentan
con
strucciones de conjuntos Bh [g] sobre los grupos abelianos Fh , + , Zd , + , y
(Zm1 × · · · × Zmd , +), para d ≥ 2, h ≥ 2, y g ≥ 1, con F cualquier campo.
Palabras y frases clave. Conjuntos de Sidon, Conjuntos Bh .
1. Introduction
Let g and h denote positive integers with h ≥ 2. Let G be an abelian additive
group denoted by (G, +). The set A = {a1 , . . . , ak } ⊆ G is a Bh [g] set on G if
every element of G can be written in at most g ways as sum of h elements in
A, that is, if given x ∈ G, the solutions of the equation x = a1 + · · · + ah , with
a1 , . . . , ah ∈ A, are at most g (up to rearrangement of summands). If g = 1, A
is a Bh set, while if g = 1 and h = 2, A is a Sidon set.
Let Fh (G, g) denote the largest cardinality of a Bh [g] on G. If g = 1 we
write Fh (G). Furthermore, if G is the direct product of d ≥ 2 abelian groups
and A is a Bh [g] set on G, sometimes we say that A is a d−dimensional Bh [g]
set on G. For N ∈ N, let [0, N − 1] := {0, 1, . . . , N − 1}. If Zd denotes the set of
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DIEGO RUIZ & CARLOS TRUJILLO
all d−tuples of integer numbers and [0, N − 1]d denotes the cartesian product
of [0, N − 1] with itself d times, we define
Fhd (N, g) := max{|A| : A ⊆ [0, N − 1]d , A ∈ Bh [g]}.
The main problem on Bh [g] sets consists on establishing the largest cardinality
of a Bh [g] set on a finite group G. With analytical constructions it is possible to
characterize lower bounds for Fh (G, g), while using counting and combinatorial
techniques, it is possible to characterize upper bounds. In this work we focus on
constructions
known lower bounds for Fh (G, g) on particular groups
to obtain
G ( Fh , + , Zd , + , and (Zm1 × · · · × Zmd , +) for any field F and d ≥ 2, h ≥ 2,
g ≥ 1), while other works are focused on upper bounds [1], [4], [11].
Different works have introduced constructions of Bh [g] sets for particular
values of h, and g. On (Z, +), the most obvious construction of Sidon sets is
given by Mian–Chowla using the greedy algorithm [2]. This result is generalized
by O’Bryant for any h ≥ 2 and any g ≥ 1 in [10].
Other constructions of Bh sets are due to Rusza, Bose, Singer,
and Erdös
& Turán. Rusza constructs a Sidon set on the group Z(p2 −p) , + for p prime.
Bose’s construction initially consider h = 2 but could
be generalized for any
h ≥ 2 and any prime power q on the group Zqh −1 , + . Similarlyto Bose, Singer
constructs a Bh set with q + 1 elements on Z(qh+1 −1)/(q−1) , + . Actually this
construction can be established using Bose’s construction [8]. Finally, based on
quadratic residues modulo a fixed prime p, Erdös & Turán construct Sidon sets
on (Z, +) [10].
In dimension d = 2 some constructions are due to Welch, Lempel, Golomb
[6], Trujillo [12], and C. Gómez & Trujillo [8]. Welch constructs Sidon sets
with p − 1 elements on the groups (Zp−1 × Zp , +), (Zp × Zp−1 , +), generalized
in [7] to the groups (Zq−1 × Fq , +) and (Fq × Zq−1 , +), respectively, where Fq
is the finite field with q elements. Golomb constructs Sidon sets with q − 2
elements on the group (Zq−1 × Zq−1 , +) (Lempel’s construction is a particular
case of Golomb). Trujillo in [12] presents an algorithm to construct Sidon sets
on (Z × Z, +) from a given Sidon set on
(Z, +). Finally, C. Gómez & Trujillo
construct Bh sets on Zp × Zph−1 −1 , + [8].
In higher dimensions, Cilleruelo in [4] presents a way of mapping Sidon sets
in N to Sidon sets in Nd for d ≥ 2, from which is possible to obtain a relation
between the functions Fh (N d ) and Fhd (N ).
In this work we present constructions of d−dimensional Bh [g] sets (d ≥ 2)
on special abelian groups. The first construction uses the elementary symmetric polynomials and the Newton’s identities to generalize a construction done
initially for d = 2 [3]. In the second construction we generalize Trujillos’s algorithm given in [12] to any dimension d and all h ≥ 2, g ≥ 1, obtaining lower
bounds for Fhd (N, g) from a known lower bounds for Fh (N d , g). Finally, using
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CONSTRUCTION OF BH [G] SETS IN PRODUCT OF GROUPS
167
a homomorphism between abelian groups, we construct d−dimensional Bh [g 0 ]
sets from d−dimensional Bh [g] sets, with g a divisor of g 0 .
The remainder of this work is organized as follows:For any finite field F, Section 2 describes a construction of Bh sets on Fh , + , where Fh denotes the set
of all h−tuples
of elements of F. Section 3 presents a construction of Bh [g] sets
on Zd , + , and in Section 4 we construct Bh [g] sets on (Zm1 × · · · × Zmd , +).
Furthermore, we present a generalization of a Golomb Costas array construction. Finally, Section 5 describes the concluding remarks of this work.
2. Construction of Bh sets on Fh , +
Let p be a prime number. Note that A := {(x, x2 ) : x ∈ Zp } is a B2 set on
(Zp × Zp , +) [3]. In this section we generalize this construction using h−tuples
(h > 2). First we introduce the following notations and definitions.
Let n be a positive integer. The elementary symmetric polynomials in the
variables x1 , . . . , xn , written by σk (x1 , . . . , xn ) for k = 1, . . . , n, is defined as
X
σk (x1 , . . . , xn ) :=
xj1 · · · xjn .
1≤j1 <···<jk ≤n
If k = 0 we consider σ0 (x1 , . . . , xn ) = 1. For n = 3 we have
σ0 (x1 , x2 , x3 ) = 1,
σ1 (x1 , x2 , x3 ) = x1 + x2 + x3 ,
σ2 (x1 , x2 , x3 ) = x1 x2 + x1 x3 + x2 x3 ,
σ3 (x1 , x2 , x3 ) = x1 x2 x3 .
Note that the elementary symmetric polynomials appear in the expansion of a
linear factorization of a monic polynomial
n
Y
(λ − xj ) =
j=1
n
X
(−1)k σk (x1 , . . . , xn )λn−k .
k=0
Note also that if pk (x1 , . . . , xn ) = xk1 + · · · + xkn , the Newton’s identities are
given by
kσk (x1 , . . . , xn ) =
k
X
(−1)i−1 σk−i (x1 , . . . , xn )pi (x1 , . . . , xn ),
(1)
i=1
for each 1 ≤ k ≤ n and for an arbitrary number n of variables.
Theorem 2.1. Let F be a field with characteristic zero or p > h. The set
A := {(x, x2 , . . . , xh ) : x ∈ F},
is a Bh set on Fh , + .
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DIEGO RUIZ & CARLOS TRUJILLO
Proof. Let s ∈ Fh . Suppose there exist two different representations of s as
sum of h elements of A as follows
s = (a1 , . . . , ah1 ) + · · · + (ah , . . . , ahh ) = (b1 , . . . , bh1 ) + · · · + (bh , . . . , bhh ),
Ph
Ph
ai , bi ∈ F for i = 1, . . . , h. Note that for all k = 1, . . . , h, i=1 aki = i=1 bki .
Ph
P
h
k
k
Because pk (a1 , . . . , ah ) =
i=1 ai and pk (b1 , . . . , bh ) =
i=1 bi , using (1)
recursively we have σi (a1 , . . . , ah ) = σi (b1 , . . . , bn ), for all i = 1, . . . , h, that is
a1 + · · · + ah = b1 + · · · + bh ,
a1 a2 + · · · + ah−1 ah = b1 b2 + · · · + bh−1 bh ,
···
a1 . . . ah = b1 . . . bh ,
which implies that the elements of the sets {a1 , . . . , ah } and {b1 , . . . , bh } are
roots of the same polynomial q(x) on F[x], i.e.,
q(x) = (x − a1 ) · · · (x − ah ) = (x − b1 ) · · · (x − bh ).
That is, {a1 , . . . , ah } = {b1 , . . . , bh } (F[x] is a unique factorization domain).
Thus, cannot be possible to have two different representations
of s ∈ F as sum
X
of h elements of Fh and A is a Bh set on Fh , + .
n
Consider the case when F is the finite field Fq , with
q = p for some n ∈ N
n
and p prime. Note that the groups (Fpn , +) and Fp , + are isomorphic, because
if θ is a root of an irreducible polynomial of degree n over Fp in an extension
field, the function
φ:
Fpn
a0 + · · · + an−1 θn−1
→
7
→
Fnp
(a0 , . . . , an−1 )
(2)
defines an isomorphism between them.
Corollary 2.2. For all p >h prime and for all n ∈ N there exists a Bh set
with pn elements on Zhn
p ,+ .
Proof. It follows immediately from Theorem 2.1 and the isomorphism φ given
X
in (2).
We illustrate these results in the following example.
Example 2.3. Consider h = n = 2 and p = 3. Let p(x) = x2 + 1 be an
irreducible polynomial on Z3 . Suppose that θ is a root of p(x) in an extension
field of Z3 . The field with 9 elements is given by
F9 = {a + bθ : a, b ∈ Z3 }
= {0, 1, 2, θ, θ + 1, θ + 2, 2θ, 2θ + 1, 2θ + 2}.
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Using Theorem 2.1 we know that
A=
(0, 0), (1, 1), (2, 1), (θ, 2), (θ + 1, 2θ), (θ + 2, θ),
(2θ, 2), (2θ + 1, θ), (2θ + 2, 2θ)
is a Sidon set on (F9 × F9 , +) = F29 , + . Furthermore, using Corollary 2.2 we
have
(0, 0, 0, 0), (0, 1, 0, 1), (0, 2, 0, 1), (1, 0, 0, 2), (1, 1, 2, 0),
B=
(1, 2, 1, 0), (2, 0, 0, 2), (2, 1, 1, 0), (2, 2, 2, 0)
is a Sidon set on (Z3 × Z3 × Z3 × Z3 , +) = Z43 , + .
3. Construction of Bh [g] sets on Zd , +
In this section we present a construction of Bh [g] sets for all h, g ≥ 2 on
Zd , + . This construction generalizes a construction introduced by Trujillo
in [12] which allows to obtain Sidon sets on (Z × Z, +) from a Sidon set on
(Z, +). Our generalization also allows to construct d−dimensional Bh [g] sets
for all h, g ≥ 2 and any dimension d, from which it is possible
to determine a
way to map Bh [g] sets on (Z, +) into Bh [g] sets on Zd , + .
Let d, N be positive integers greater than 1. Let A denote a subset of Z+ . If
a ∈ A, [a]N = (nk , . . . , n1 , n0 )N represents the integer a = nk N k +· · ·+n1 N +n0
in base N notation, where k is a nonnegative integer and 0 ≤ nj ≤ N − 1, for
j = 0, 1, . . . , k. We denote the set obtained from the representation of each
element of A in base N as [A]N . Because every positive integer can be written
uniquely in base N , then
|A| = |[A]N |.
Note that if A ⊆ 0, N d − 1 , then [A]N ⊆ [0, N − 1]d .
Theorem 3.1. If A is a Bh [g] set contained in [0, N d − 1], then [A]N is a
Bh [g] set contained in [0, N − 1]d .
Proof. Let s be a d−tuple in Zd obtained as sum of h elements in [A]N .
Suppose there exist g + 1 representations of s as follows
s = [a1,1 ]N + · · · + [a1,h ]N = · · · = [ag+1,1 ]N + · · · + [ag+1,h ]N ,
(3)
where ai,j ∈ A for all 1 ≤ i ≤ g + 1, 1 ≤ j ≤ h. Consider the
representation of
each ai,j ∈ A in base N as [ai,j ]N = n(d−1,i,j) , . . . , n(0,i,j) . Note that for any
1≤i≤g+1
[ai,1 ]N + · · · + [ai,h ]N = n(d−1,i,1) , . . . , n(0,i,1) + · · · + n(d−1,i,h) , . . . , n(0,i,h)
= n(d−1,i,1) + · · · + n(d−1,i,h) , . . . , n(0,i,1) + · · · + n(0,i,h) .
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DIEGO RUIZ & CARLOS TRUJILLO
Furthermore
(n(d−1,i,1) +· · ·+n(d−1,i,h) )N d−1 +· · ·+(n(0,i,1) +· · ·+n(0,i,h) ) = ai,1 +· · ·+ai,h
which implies from (3) that
a1,1 + · · · + a1,h = · · · = ag+1,1 + · · · + ag+1,h .
(4)
Because A is a Bh [g] set, using (4) we know there exist `, m with ` 6= m and
1 ≤ `, m ≤ g + 1, such that
{a`,1 , . . . , a`,h } = {am,1 , . . . , am,h }.
Since representation in base N notation is unique we have
{[a`,1 ]N , . . . , [a`,h ]N } = {[am,1 ]N , . . . , [am,h ]N },
That is, it is not possible to have g +1 representations of s as sum of helements
X
of A. Therefore [A]N is a Bh [g] set contained in [0, N − 1]d ⊂ Zd , + .
Example 3.2. Note that A = {1, 2, 7} is a Sidon set on (Z8 , +). In [12] Trujillo
constructs a B2 [2] set on (Z, +) as follows
B := A ∪ (A + m) ∪ (A + 3m) = {1, 2, 7, 9, 10, 15, 25, 26, 31},
with m = 8. Because B ⊆ [0, 25 − 1], using Theorem 3.1 we have that
(0, 0, 0, 0, 1), (0, 0, 0, 1, 0), (0, 0, 1, 1, 1), (0, 1, 0, 0, 1), (0, 1, 0, 1, 0),
[B]2 =
(0, 1, 1, 1, 1), (1, 1, 0, 0, 1), (1, 1, 0, 1, 0), (1, 1, 1, 1, 1)
is a B2 [2] set contained in [0, 1]5 . Note also that B ⊆ [0, 62 − 1], so
[B]6 = {(0, 1), (0, 2), (1, 1), (1, 3), (1, 4), (2, 3), (4, 1), (4, 2), (5, 1)}
is a B2 [2] set contained in [0, 5]2 .
4. Construction of Bh [g] sets on (Zm1 × · · · × Zmd , +)
This section extend a construction of Bh [g] sets given in [5] for h = 2 and d = 1,
to all h ≥ 2 and any dimension d > 1. First, we introduce the following result.
Lemma 4.1. Let G and G0 be two abelian groups and let φ : G → G0 define
a homomorphism. If A is a Bh [g] set on G and |Ker(φ)| = g 0 , then φ(A) is a
Bh [gg 0 ] set on φ(G), where gg 0 denotes the product between g and g 0 .
The proof is given in [9].
Now, let m1 , . . . , md and g1 , . . . , gd be positive integers. Using Lemma 4.1
we have the following result.
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Theorem 4.2. Let A be a Bh [g] set on (Zm1 × · · · × Zmd , +). If g1 , . . . , gd are
divisors of m1 , . . . , md , respectively, then
md
m1
: (a1 , . . . , ad ) ∈ A
B :=
a1 mod
, . . . , ad mod
g1
gd
is a Bh [gg1 · · · gd ] set on Z mg 1 × · · · × Z md , + .
gd
1
Proof. Using notation used in Lemma 4.1, let G = (Zm1 × · · · × Zmd , +) and
Z mg 1 × · · · × Z md , + and define the homomorphism φ : G → G0 as
gd
1
md
1
φ(b1 , . . . , bd ) = b1 mod m
,
.
.
.
,
b
mod
d
g1
gd . We establish Ker(φ) as follows.
G0 =
Note that (b1 , . . . , bn ) ∈ Ker(φ) if and only if φ(b1 , . . . , bn ) = (0, . . . , 0), that
is, if
m1
md
b1 mod
, . . . , bd mod
= (0, . . . , 0).
g1
gd
mi
i
Note also that bi mod m
gi = 0 if and only if bi = ki gi , for ki ∈ [1, gi ] and for
mi
all i = 1, . . . , d, which implies that bi mod gi = 0 in exactly gi values. Thus,
Qd
|Ker(φ)| = i=1 gi .Finally, using Lemma
4.1 we have that B = φ(A) is a
X
Bh [gg1 · · · gd ] set on Z mg 1 × · · · × Z md , + .
1
gd
Given q a prime power and F a field, to illustrate Theorem 4.2 we present a
construction of Sidon sets on (Zq−1 × Zq−1 , +), which is based on the discrete
logarithm1 on Fq .
Proposition 4.3. Let q = pn a prime power. If α, β are primitive elements of
F∗q and a ∈ F∗q , then
G(α, β, a) := {(i, logβ (a − αi )) : i = 1, . . . , q − 1, αi 6= a}
(5)
is a Sidon set on (Zq−1 × Zq−1 , +).
Proof. Suppose there exist u, v, w, y ∈ G(α, β, a) such that u + v = w + y.
Using (5) we know that there exist i, j, k, ` ∈ [1, q − 1] such that
(i, logβ (a − αi )) + (j, logβ (a − αj )) = (k, logβ (a − αk )) + (`, logβ (a − α` )) (6)
where αi , αj , αk , α` are not equal to a. From (6) we have
(i + j) ≡ (k + `) mod (q − 1),
i
logβ (a − α ) + logβ (a − αj ) ≡ (logβ (a − αk ) + logβ (a − α` )) mod (q − 1),
1 If θ is a primitive of F , log (x) denotes the unique integer k ∈ [1, q − 1] such that θ k = x
q
θ
on Fq .
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DIEGO RUIZ & CARLOS TRUJILLO
what implies that (a − αi )(a − αj ) = (a − αk )(a − α` ). We have in F∗q
αi αj = αk α` ,
αi + αj = αk + α` ,
that is, αi , αj , and αk , α` are roots of a polynomial q(x) ∈ F[x] of degree 2 (i.e.,
q(x) = (x + αi )(x + αj ) = (x + αk )(x + α` )). Therefore, {αi , αj } = {αk , α` } and
{i, j} = {k, `}, which implies that is not possible to have two representations
of an element in Zq−1 × Zq−1 as sum of two elements of G(α, β, a). That is,
X
G(α, β, a) is a Sidon set on (Zq−1 × Zq−1 , +).
Example 4.4. First we apply Proposition 4.3 to construct a Sidon set on
hZ16 × Z16 , +i. Let q = p = 17, and let α = 3, β = 5 be primitive elements of
Z∗17 . With a = 1
(1, 14), (2, 10), (3, 2), (4, 1), (5, 4), (6, 13), (7, 15), (8, 6),
G(3, 5, 1) =
(9, 12), (10, 7), (11, 11), (12, 5), (13, 3), (14, 8), (15, 9)
is a Sidon set on (Z16 × Z16 , +). Now, if g1 = g2 = 2, using Theorem 4.2,
(1, 6), (2, 2), (3, 2), (4, 1), (5, 4), (6, 5), (7, 7), (0, 6),
A=
(1, 4), (2, 7), (3, 3), (4, 5), (5, 3), (6, 0), (7, 1)
is a B2 [4] set on (Z8 × Z8 , +).
5. Concluding remarks
Using the constructions given in this work we can obtain lower bounds and
closed formulas for Fhd (G, g), for some abelian group G and some values of d, h
and g.
Note from Theorem 2.1 and Corollary 2.2 that F2h (Fhq ) ≥ q for q a prime
power. Particularly if h = 2 and q = p prime we have F22 (Zp × Zp ) ≥ p, but
it is easy to establish that F22 (Zp × Zp ) = p [7]. A natural question to state
is the following: Can we obtain a similar result, as the last one, on the group
(Zp × Zp × Zp , +)? That is,
F23 (Zp × Zp × Zp ) ∼ p3/2 ?
Now, using Theorem 3.1, for integers d, g, N ≥ 1 and h ≥ 2 we know that
Fh (N d , g) ≤ Fhd (N, g)
Particularly, if d = 2, h = 2, and g = 1 we have that F21 (N 2 ) ≤ F22 (N ), which
implies that good constructions of Sidon sets on Z give good lower bounds for
Sidon sets on Z × Z. Furthermore, an interesting work consists in to analyze
the behavior of the difference F22 (N ) − F21 (N 2 ) when N grows.
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CONSTRUCTION OF BH [G] SETS IN PRODUCT OF GROUPS
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Finally, from Proposition 4.3 we can establish that F22 (Zq−1 ×Zq−1 ) ≥ q −2,
which lead us to wonder if is it possible to state that F22 (Zq−1 × Zq−1 ) = q − 1?
Acknowledgment. We thank to COLCIENCIAS and Universidad del Cauca
for the support to our research group “Algebra, Teorı́a de Números y Aplicaciones–ALTENUA ERM” under the projects 110356935047 and VRI–3744. We
also thank to the anonymous referee for their helpful comments. This work is
dedicated to the memory of Javier Cilleruelo (1961-2016).
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DIEGO RUIZ & CARLOS TRUJILLO
[12] C. Trujillo, Sucesiones de sidon, Ph.D. thesis, Universidad Politécnica de
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(Recibido en abril de 2016. Aceptado en julio de 2016)
Departamento de Matemáticas
Universidad del Cauca
Calle 5 No. 4-70
Popayán, Colombia
e-mail: [email protected]
e-mail: [email protected]
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